A188924 Decimal expansion of sqrt(4+sqrt(15)).

2, 8, 0, 5, 8, 8, 3, 7, 0, 1, 4, 7, 5, 7, 7, 8, 7, 1, 5, 0, 9, 8, 0, 8, 8, 8, 0, 9, 5, 6, 9, 3, 0, 4, 9, 6, 2, 8, 4, 2, 7, 5, 1, 3, 0, 9, 9, 9, 0, 9, 4, 3, 4, 7, 7, 6, 4, 5, 0, 9, 8, 7, 1, 0, 0, 2, 1, 7, 7, 7, 4, 0, 8, 0, 4, 8, 2, 7, 6, 6, 2, 3, 9, 4, 2, 0, 5, 3, 7, 7, 0, 7, 4, 1, 9, 7, 0, 2, 6, 5, 0, 0, 2, 9, 7, 0, 9, 4, 2, 6, 8, 9, 7, 2, 7, 1, 2, 2, 1, 3, 6, 7, 0, 3, 8, 6, 0, 7, 4, 5

Offset: 1

Author

Clark Kimberling, Apr 13 2011

Keywords

• nonn
• cons

Comments

Decimal expansion of the length/width ratio of a sqrt(6)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A sqrt(6)-extension rectangle matches the continued fraction [2,1,4,6,1,1,2,25,3,...] for the shape L/W=sqrt(4+sqrt(15)). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...]. Specifically, for the sqrt(6)-extension rectangle, 2 squares are removed first, then 1 square, then 4 squares, then 6 squares,..., so that the original rectangle of shape sqrt(4+sqrt(15)) is partitioned into an infinite collection of squares.

Examples

			2.8058837014757787150980888095693049628427513...
		

Crossrefs

Cf. A020797, A115754, A188925.

Programs

Mathematica

r = 6^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

PARI

sqrt(3/2) + sqrt(5/2) \\ Hugo Pfoertner, Feb 20 2024

Formula

Equals A115754 + 10*A020797. - Hugo Pfoertner, Feb 20 2024